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Neutral and interordinal intervals in MOS scales: Difference between revisions

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* To show that these actually occur in bL(a-b)s, consider smaller and larger j-steps ( 1 ≤ j ≤ a-1) in the parent mos. These intervals also occur in the mos aLbs separated by s, and the number of j’s (“junctures”) that correspond to these places in aLbs is exactly a-1. These j's correspond to values of k such that larger k-step < smaller k+1-step. Note that we are considering “junctures” between k-steps and k+1-steps in aLbs, excluding k = 0 and k = a+b-1, so the total number of “junctures” to consider is finite, namely a+b-2. This proves part (1).
* To show that these actually occur in bL(a-b)s, consider smaller and larger j-steps ( 1 ≤ j ≤ a-1) in the parent mos. These intervals also occur in the mos aLbs separated by s, and the number of j’s (“junctures”) that correspond to these places in aLbs is exactly a-1. These j's correspond to values of k such that larger k-step < smaller k+1-step. Note that we are considering “junctures” between k-steps and k+1-steps in aLbs, excluding k = 0 and k = a+b-1, so the total number of “junctures” to consider is finite, namely a+b-2. This proves part (1).


Part (2) is easier to see: when larger k-step = smaller k+1-step, larger k+1-step - smaller k-step = 2(L-s) = 2s = L. The step L is 4 steps in 2n-edo.
Now part (2) is almost immediate: when larger k-step = smaller k+1-step, larger k+1-step - smaller k-step = 2(L-s) = 2s = L. The step L is 4 steps in 2n-edo.
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